Find the Roots (Zeros) -x=-6x^2+1

$- x = - 6 x^{2} + 1$

Step 1

Add $6 x^{2}$ to both sides of the equation.

$- x + 6 x^{2} = 1$

Step 2

Subtract $1$ from both sides of the equation.

$- x + 6 x^{2} - 1 = 0$

Step 3

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$- u + 6 u^{2} - 1 = 0$

Factor by grouping.

Reorder terms.

$6 u^{2} - u - 1 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 1 = - 6$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- u$ .

$6 u^{2} - (u) - 1 = 0$

Rewrite $- 1$ as $2$ plus $- 3$

$6 u^{2} + (2 - 3) u - 1 = 0$

Apply the distributive property.

$6 u^{2} + 2 u - 3 u - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 u^{2} + 2 u) - 3 u - 1 = 0$

Factor out the greatest common factor (GCF) from each group.

$2 u (3 u + 1) - (3 u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(3 u + 1) (2 u - 1) = 0$

Replace all occurrences of $u$ with $x$ .

$(3 x + 1) (2 x - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 1 = 0$

$2 x - 1 = 0$

Step 5

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 6

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 7

The final solution is all the values that make $(3 x + 1) (2 x - 1) = 0$ true.

$x = - \frac{1}{3}, \frac{1}{2}$

Step 8

Find the Roots (Zeros) -x=-6x^2+1

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